Evaluate the integral: I = integral dx/(x ln^5 x)

Evaluate the integral: $$ I = \int \frac{dx}{x \ln^5 x}. $$

Explanation of Method

The substitution $u=\ln x$ converts $\frac{dx}{x}$ into $du$, turning the integrand into a simple negative power of $u$, which is handled directly by the power rule.

Substitute $u=\ln x$.
Integrate the resulting negative power of $u$.
Substitute back and add the constant of integration.

Step 1 · Substitution

We use the substitution:

$$ u = \ln x \quad \Rightarrow \quad du = \frac{dx}{x}. $$

Rewriting the integral in terms of $u$:

$$ I = \int \frac{du}{u^5}. $$

Step 2 · Integrate

Since $\dfrac{1}{u^5}=u^{-5}$, we integrate:

$$ I = \int u^{-5}\,du. $$

Using the power rule:

$$ I = \frac{u^{-4}}{-4} = -\frac{1}{4u^4} + C. $$

Step 3 · Substitute back

Substituting back $u = \ln x$:

$$ I = -\frac{1}{4(\ln x)^4} + C. $$

Summary Table

Quantity	Value
Substitution	$u = \ln x$
$dx/x$ in terms of $du$	$du$
Antiderivative	$-\tfrac{1}{4u^4}$

Additional Notes

The domain of $I$ requires $x>0$ and $x\neq 1$, since $\ln x$ must be defined and nonzero (as it appears raised to a negative power).
This substitution generalizes to any integrand of the form $\dfrac{f(\ln x)}{x}$: the factor $\dfrac{dx}{x}$ always converts cleanly into $du$ when $u=\ln x$.
The result can be checked by differentiating $-\dfrac{1}{4(\ln x)^4}$ with the chain rule and confirming it returns $\dfrac{1}{x\ln^5x}$.

Result

The solution to the integral is:

$$ I = -\frac{1}{4(\ln x)^4} + C. $$